"You're going to have to be more specific"
- Question: "... what is the function of be going to in this case?"
It's the construction to be + going + infinitive
This usage refers to the near future as opposed to a Future Indefinite tense. In other words, your sentence, "You're going to have to be more specific" can be rewritten as "You will have to be more specific".
This is correct and is commonly used in conversations, replacing Future tense.
- Is it a polite way to say one should be more specific?
Well, for sure, it tones it down. "You should be more specific" sounds straight and abrupt.
Future
(Necessary
(you, specific (you)) -- in the future, it will be necessary for you to be more specific.